To what extent fuzzy set theory and structural equation modelling can
measure functionings? An application to child well being
Tindara Addabbo,* Maria Laura Di Tommaso**and Gisella Facchinetti***
* CAPP and CHILD Department of Economics,
Faculty of Economics “Marco Biagi”,University of Modena and Reggio Emilia
[email protected]
** CHILD and Department of Economics “Cognetti de Martiis”
University of Turin
[email protected]
*** Department of Economics, Faculty of Economics “Marco Biagi”
University of Modena and Reggio Emilia
[email protected]
Materiali di Discussione del Dipartimento di Economia Politica n.468 Settembre 2004
JEL: C1, C6, D6, D3
1
Abstract
This paper explores the possibilities of using fuzzy inference system and
structural equation modelling to measure capabilities both at a theoretical and
empirical level.
Fuzzy set theory has been already used to measure functionings (Chiappero
Martinetti 1996, 2000, Lelli 2001) while structural equation modelling has not
been used till now (apart from some preliminary results on children well being in
India presented by Di Tommaso 2003). In this paper we outline the major
advantages and disadvantages of both the approaches both in terms of the
statistical assumptions implied and in terms of their ability to measure
functionings. Are the statistical assumptions implied by these approaches
compatible with the capability approach? What limitations the statistical
assumptions impose to the capability approach?
In order to assess to what extent these two statistical techniques work, we will
apply them to measure child well being with a capabilities approach. The aim is
also how to propose a list of capabilities with reference to children well being in
Italy.
The applied part of the paper will use a data set based on a ISTAT (Italian
National Statistical Office) multipurpose survey on family and on children
condition in Italy to recover information on children’s education, the sociodemographic structure of their families, child care provided by relatives and
parents according to the type of activities in which the children are involved.
In the conclusion of the paper, we outline if and to what exte nt these statistical
and fuzzy techniques can be used to measure functionings with special
reference to child well being.
2
Introduction1
This paper explores the possibilities of using fuzzy inference system and
structural equation modelling to measure capabilities both at a theoretical and
empirical level.
Fuzzy set theory has been already used to measure functionings (Chiappero
Martinetti 1996, 2000, Lelli 2001) while structural equation modelling has not
been used till now (apart from some preliminary results on children well being in
India presented by Di Tommaso 2003). In this paper we outline the major
advantages and disadvantages of both the approaches both in terms of the
statistical assumptions implied and in terms of their ability to measure
functionings. Are the statistical assumptions implied by these approaches
compatible with the capability approach? What limitations the statistical
assumptions impose to the capability approach?
In order to assess to what extent these two statistical techniques work, we will
apply them to measure child well being with a capabilities approach. The aim is
also how to propose a list of capabilities with reference to children well being in
Italy this is attempted in the first part of the paper, whe reas fuzzy set theory and
structural equation modelling are introduced in Secion 2.1 and 2.2.
The applied part of the paper will use a data set based on a ISTAT (Italian
National Statistical Office) multipurpose survey on family and on children
condition in Italy to recover information on children’s education, the sociodemographic structure of their families, child care provided by relatives and
parents according to the type of activities in which the children are involved.
In the conclusion of the paper, we outline if and to what extent these statistical
and fuzzy techniques can be used to measure functionings with special
reference to child well being.
Section I Capability approach and child well being
Endorsing a specific list of capabilities in the case of children well being in
Western societies and in particular in Italy it is very different from endorsing a
list for adults. In what follows we explain why and in which way.
Nussbaum (2003) argues that the capabilities approach should not only include
the capabilities of the people who are in need (typically children or elderly) but
the capability approach should endorse a theory of social justice where the
th
1
We would like to thank participants to the 4 International Conference on the Capability
approach (Enhancing Human Security, University of Pavia, Italy) and to the 2004 IAFFE
Conference (Oxford, UK ) for their stimulating comments on a previous version of this paper. The
responsibility of what follows is entirely to be attributed to the authors.
3
subjects are not anymore only “fully cooperating members of society over a
complete life” 2 .
“ So I believe we need to delve deeper, redesigning the political conception of
the person, bringing the rational and the animal into a more intimate relation
with one another, and acknowledging that there are many types of dignity in the
world, including the dignity of mentally disabled children and adults, the dignity
of the senile demented elderly, and the dignity of babies at the breast.
…………………………………………………………
We thus need to adopt a political conception of the person that is more an
Aristotelian than Kantian, one that sees the person from the start as both
capable and needy – “ in need of a rich plurality of life-activities “ to use a
Marx’s phrase, whose availability will be the measure of well-being.” 3
In order therefore to conceptualise children well-being in Italy we consider
children as subjects. In the language of rights, children rights have been
established for a long time. But in the case of capabilities we are not aware of
any conceptualisation.
Another theoretical problem is to assess to what extent what we observe is the
result of the child’s choice (given a vector of commodities and his/her
capabilities) or it is determined by parents or other individuals, to what extent
the functionings we observe for a given capability are just affected by parents’
and social environment or are a projection of parents’ ideas and of actual
constraints. For instance to what extent observing that a child is attending a
piano lecture is a result of his/her choice given his/her capability or it is
determined by the wish of his/her parents to play piano and to have a child
doing it? Probably having longitudinal data or data on the child achievements in
each activities could provide us a better signal on whether the observed
functioning is just a realization of parents’ wishes or is an expression of child’s
capability (even if this is definitely connected to parents’ wishes and
expectations).
Moreover children’s capabilities are bound to be affected not only in a static way
by family, personal, social and environmental factors but the relationship with
child’s environment (including an extended net of relationship and of institutional
constraints like for instance the availability of kindergarten or of full-time schools
and the very quality of the school system) is bound to affect the very
development of capabilities on the individual capability profile over his/her life
cycle as the child ages. For instance if one takes the attendance of child care
services its impact on children’s cognitive outcomes can be observed over
child’s life cycle.4
Different environmental factors may interact in affecting child’s capabilities or
their conversion in functionings. Again taking the example of childcare
attendance empirical analyses show that childcare attendance at early age
2
Rawls 1980, pag 546, citation taken from Nussbaum 2003.
Nussbaum 2003 pag 29-30.
4
Walfogel (2002) provides a survey on the impact of childcare and mother’s employment status
on child outcomes.
3
4
produces different effects on child’s cognitive ability (and therefore may
differently affect child’s education and knowledge capability) according to family
characteristics. 5
In this regard we believe that an interdisciplinary group with developmental
psychologists, child psychoanalysts, pedagogists can improve our
understanding of the links amongst capabilities over the life cycle and between
environmental factors and capabilities in a dynamic perspective. This is actually
an aim that we would like to pursue in the future application of our analysis, but
even if we cannot trace now a theoretically consistent net of relationship
between perspective capabilities, actual capabilities and functionings we are
aware of the importance that for instance lack of social interaction can play in
the developing of other capabilities and in their realizations for the child that we
observe now.
Section 1.1 A list of capabilities for children
In order to conceptualise children well being with a capability approach we
follow the procedure suggested by I. Robeyns (Robeyns 2003).
In her paper Robeyns provides an important methodology to endorse a list of
capabilities in order to assess gender inequality in Western countries. She
provides an exhaustive list of criteria to be implemented in order to endorse a
list of capabilities in the case of gender inequality. Robeyns’ selection criteria
are the following:
1. Explicit formulation: “the list should be explicit, discussed, and defended”
(Robeyns 2003 pp70)
2. Methodological justification: in drawing a list we should justify the
method used.
3. Sensitivity to contest: the level of abstraction of the list should be
appropriate for the aim of the research.
4. Different levels of generality: there are 2 levels: the first is the ideal list
that fulfils the above criteria; the second is the empirically implementable
list.
5. Exhaustion and non reduction: the list of capabilities should include all
important elements.
In this paper we argue that also in the case of children well being, it is important
to use that methodology in order to provide a general frame where the specific
measures of child well being can be set against.
5
Waldfogel (2002,p.530) refers to literature showing that ‘children from families that are
economically disadvantaged gain more from child care in terms of their cognitive development
that do other children’ and also to NICHD study in the USA that shows that ‘infants whose
parents had more sensitive childrearing styles fared better than other children in early child
care’.
5
The criterium of explicit formulation.
1. Life and physical health; being able to ha ve good health, not dying
prematurely, to be adequately nourished; to have adequate shelter.
This capability is very much related to family income (and the possibility
of having private health care) and to the quality of the national health
service across regions,6
In Western society in general the problem is not malnutrition or stunting
but unhealthy nutritional habits leading to obesity and overweight.
2. Mental Well being: the capability to have a mental health can be applied
to children. Do children ha ve adequate psychological support when
needed? Is children psychological development taken into account in the
schools?
3. Bodily Integrity. “Being able to move freely from place to place; to be
secure against violent assault, including sexual assault and domestic
violence.”
The capability of moving freely in the territory is often denied to Italian
children, specially to those who live in towns: there has been an
increasing concern among parents to let children go to school alone, to
play in courtyards and in playgrounds by themselves. This is due to the
increase of criminality rates. Security for children in terms of domestic
violence and sexual assault are also important issues.
Sexual and domestic violence are also an important capability for
children. Beating children is not so morally condemned in Italy as it is in
Northern European countries.
In the above issues there is a gender components that we need to take
into account.
4. Social interaction: Is the child able to interact with other people? What
people does the child interact with? Italy is characterized by one of the
lowest fertility rate in the world and the probability to observe a child
interacting with sisters or brothers or with cousins is not so high. How are
the functionings connected to this capability connected to different
(extended) family composition, or with the possibility to attend schools or
6
The latter has a great degree of variation across regions, a signal of different qualities in the
health service is the observed mobility of patients across regions in Italian hospitals. In a system
of indicators affecting this capability one should include data on quality of national health
service, presence of special services devoted to children and on the ratio between
pediatrician/children living in one region.
6
kindergarten. 7 How is the possibility of interacting with parents affected
by their labour market condition, by their time constraints, by their
educational level, wishing and capability to interact with their children.
5. Education and knowledge: being able to be educated and to use and
produce knowledge. This is a basic capability for the development of
children. How does the quality of Italian schools differ across the
regions? The Innocenti report (Istituto degli Innocenti, 2002 ) shows how
the availability of kindergartens varies across regions. The drop off rates
in some regions is also very high for teenagers.
6.
Leisure activities, play. The role of this capability in children’s well being
is essential, however even its functionings are not easily observable. We
can look for estimates of consumption of toys by using expenditure
surveys, or look for time use data. Asking how this capability can be
affected by the environment we could look for data on the presence of
schools or play centres for children in different age groups, or of
laboratories dedicated to children (painting, music, dance…) and on their
attendance and let these data interact with available individual data on
the actual leisure activities of children. Pychologists could stress that it is
not only important to assess how much the child plays but what types of
plays and activities the child does and with whom does he/she play. This
capability is strongly related to other children’s capabilities lilke social
interaction and education. Not playing alone requires interaction with
other children, parents or with other individuals who are wishing to
interact. Taking Italy we could try to assess how the functionings
connected to this capability differ across regions characterised by
different presence of schools and organized leisure activities. One could
recognize that pure leisure time not devoted to attending laboratories or
activities structured by adults is reducing for Italian children and analyse
also how new types of playing activities currently more spread amongst
them (like videogames) can affect other capabilities also in perspective.
Another important issue is to what extent the child is free from paid or unpaid work. There has been an increased concern for the amount of work
performed by children.
The criterium of methodological justification
Methodologically, we have taken into consideration the 4 different lists of
capabilities reassumed by Robeyns (2003) in table 1, and selecting the ones
that are relevant for children well being. We have excluded capabilities
concerned with political freedoms, religion, paid work, time autonomy.
7
According to Istituto degli Innocenti (2002) in year 2000 only 7,4 places were available for
every 100 children aged from 0-2 in Italy, and this percentage is the result of very different
situations across Italy from a minimum of 1,9% in Calabria to 18,3% in Emilia Romagna. Also
the possibility of attending educational services ‘integrativi’ that can offer social interaction
possibilities for children is differently spread across Italy as Istituto degli Innocenti (2002) shows.
7
The list provided in this paper is very provisional and it needs to be revised with
the help of sociologists and psychologists.
The criterium of sensitivity to context.
The list of capabilities provided above is drawn thinking to the possible
applications in social sciences; when a particular measure to promote children
well being is introduced (for instance a change in the school system), it is
relevant to understand both the effect of that measure on a single indicator of
child well being (for instance educational results) and on the compounded
measure of child well being.
The criterium of different levels of generality.
Ideally which kind of data set would be needed?
As we have stated in the first part of section 1, measuring children capabilities
would require to observe children development during their childhood. The ideal
data set would therefore follow children from their birth until they become adults
(panel data).
In fact measuring children well being is age dependent. Many functionings can
only be measured at a late stage of the child development .
We would also need detailed information for each child, family background,
income, and a detailed questionnaire for each of the above–specified
capabilities and functionings. In addition to this we would need to distinguish
between achieved functionings and capabilities. To this extent it would be
important to have information on the social environment the child leaves in.
Regional or cities data on the quality of the schooling system, the availability of
green area, sports facilities, public subsides to children recreational activities.
These data would help in measuring capabilities vis-à-vis functionings.
The data used for the empirical application of this paper are specified in Section
3.2.
The criterium of exhaustion and non-reduction.
The list of capabilities provided above includes all important elements for
measuring capabilities of Italian children and cannot be reduced.
In the empirical part of this paper we will only attempt to measure one of the
capability of the list, the social interaction.
In the applied part of this paper we will try to propose a system of observable
measures of functionings and environmental indicators relevant for the
development or being a realization of the children’s capability of social
interaction.
8
Section 2 To what extent fuzzy set theory and structural equation
modelling can measure functionings?
Section 2.1 The fuzzy model: a fuzzy expert system.
Conventional mathematics enables processing of precise information.
However, in the reality, we very often meet with imprecise information such as:
sufficient well-being, good level of quality of life, etc. People have used
imprecise information for thousands of years. However, until quite recently it
has not been used at all in methods based on conventional mathematics.
Therefore it has been lost. Because of this, the efficiency of many control,
modeling, forecasting and decision- making methods was considerably limited
all the more, as in some systems imprecise information is the only accessible
one.
The domain of mathematics dealing with imprecise information was
named Fuzzy Set Theory. This theory in connection with conventional
mathematics enables the processing and use of any information.
Since their inception in the mid 1960s by Professor Lotfi A. Zadeh, from
the University of California, (Zadeh L.A. (1965)), Fuzzy Set Theory have
triggered mixed feeling in scientific community. On one hand, a growing number
of devotees have recognized potentials of fuzzy sets to model and solve many
real problems, on the other a considerable number of opponents have fiercely
fought against these emerging tools. One of their arguments has been a lack of
applications.
The situation had change since the mid of 1980s when the so-called
“fuzzy boom” occurred primarily in Japan, but then also in Korea and Europe.
Basically, the turning point was the launching on the market of fuzzy logic
control based appliances and other equipment exemplified by subway trains,
cranes, elevators, etc.
In the same period engineering control applications were faced with a
first try to use traditional mathematics models for incomplete information
problems, such as “Expert Systems”. An Expert System is an intelligent
machine that uses knowledge and inference procedures to solve control
problems that have a so high level of complexity to require significant human
expertise for their solutions. The knowledge of an expert system consists of
facts and heuristics. The facts usually constitute a body of information that is
widely shared, publicly available, and generally agreed upon by experts in the
field. Heuristics concerns mostly private information and rules of good judgment
that characterize expert-level decision making in the field. These instruments
have had a great relevance in engineering applications, but with the increase of
problems complexity they show their limits and were abandoned.
The Japanese successful applications of fuzzy logic control, were
obtained by a “Fuzzy Evolution” of “Expert Systems”, called “Fuzzy Expert
Systems”, (Von Altrock, C. (1997)).
A Fuzzy Expert System utilizes fuzzy sets and fuzzy logic to overcome
9
some of the problems that occur when the data provided by the user are vague
or incomplete. The power of fuzzy set theory comes from the ability to describe
linguistically a particular phenomenon or process, and then to represent that
description with a small number of very flexible rules. In a Fuzzy Expert System,
the knowledge is contained both in its rules and in fuzzy sets, which hold
general description of the properties of the phenomenon under consideration.
One of the major differences between a Fuzzy Expert System and another
Expert System is that the first can infer multiple conclusions. In fact it provides
all possible solutions whose truth is above a certain threshold, and the user or
the application program can then choose the appropriate solution depending on
the particular situation. This fact adds flexibility to the system and makes it more
powerful. Fuzzy Expert Systems use fuzzy data, fuzzy rules, and fuzzy
inference, in addition to the standard ones implemented in the ordinary Expert
Systems.
Functionally a fuzzy system can be described as a function approximator.
More specifically it aims at performing an approximate implementation of an
unknown mapping f : A ⊆ R n → R m where A is a compact of Rn . By means of
variable knowledge relevant to the unknown mapping [Kosko, 1992] and [Wang,
1992] independently proved that fuzzy systems are dense in the space of
continuous functions on a compact domain and therefore can approximate
arbitrarily well any continuous function on a compact domain. The following are
the main phases of a Fuzzy Expert System design:
1. Identification of the problem and choice of the type of Fuzzy Expert
System, which best suits the problem requirement. A modular system
can be designed. It consists of several fuzzy modules linked together. A
modular approach may greatly simplify the design of the whole system,
dramatically reducing its complexity and making it more comprehensible.
2. Definition of input and output variables, their linguistic attributes (fuzzy
values) and their membership function (fuzzification of input and output).
3. Definition of the set of heuristic fuzzy rules. (IF -THEN rules).
4. Choice of the fuzzy inference method (selection of aggregation operators
for precondition and conclusion).
5. Translation of the fuzzy output in a crisp value (defuzzification methods).
6. Test of the fuzzy system prototype, drawing of the goal function between
input and output fuzzy variables, change of membership functions and
fuzzy rules if necessary, tuning of the fuzzy system, validation of results.
In building Fuzzy Expert System, the crucial steps are the fuzzification
and the construction of blocks of fuzzy rules. These steps can be handled in two
different ways. The first is accomplished by using information obtained through
interviews to the experts of the problem. The second is accomplished by using
methods of machine-learning, neural networks and genetic algorithms to learn
membership functions and fuzzy rules. The two approaches are quite different.
The first does not use the past history of the problem, but it relies on the
experience of experts who have worked in the field for years. The second is
based on past data and project into the future the same structure of the past.
We can formalize the steps in the following manner. For each linguistic
10
variable, input xi (i=1…m) and output y, we have to fix its own range of
variability Ui and V. ∀ i, (i=1…m), if ni is the number of the linguistic attribute of
the variable xi and n̂ = max ni , we define the sets
i ∈[1, m ]
A = {A1i ,A2i , ..,Aij i ,..,Ani i }, B = {B1, B2 ,...,Bk , ...,Br }
i
where ∀ j i ∈ [1, ni ], ∀ ni ∈ [1, n̂ ] A iji are the fuzzy numbers describing the
linguistic attributes of the input variable xi , and ∀k ∈ [1, r ] , Bk are the fuzzy
numbers describing the linguistic attributes of the output variable y.
At every elements of Ai and B a membership function is associated such
that
µA i (x) : U i → [0,1]
and µB k : V → [0,1]
ji
The elements of Ai and B overlap in some “grey” zone, which cannot be
characterised precisely. Many phenomena in the world do not fall clearly into
one crisp category or another. Experts, that use abstraction as a way of
simplifying the problem, can contribute to identify these “grey” zones.
The choice of the slopes of the elements of Ai and B is a mathematical
translation of what the experts think about the single terms.
The second step is the block-rules construction.
We define the set of L fuzzy rules, where
m
L ≤ ∏ ni , ∀ j i ∈ [1, ni ], ∀ ni ∈ [1, n̂ ] ∀k ∈ [1, r ]
1
IF (x1 is A ) ⊗ (x2 is. A2j ) ⊗ … ⊗ (xm is Amjm )
THEN (y is B k),
1
j1
2
(2.1-1)
(2.1-2)
The relation (2.1-1) is called “precondition” and the symbol ⊗ represents one of
the possible aggregation operators. In practical applications, the MIN and MAX
operators, or a convex combination of them, are widely used and so a
“negative” or “positive” compensation will occur between them.
Instead of Min and Max, it is also possible to use other t- norms and s-norms,
which represent different ways of linking the “and” with the “or”.
The relation (2.1-2) is called conclusion. The aggregation of precondition and
conclusion can be made in several ways. The most used are the MAX and the
BSUM methods. The choice depends on the type of application. The MAX has
the meaning of keeping as “winner” the strongest rule, in the sense that if a rule
is “firing” (activated) more than one time, the result is the maximum level of
firing. In the BSUM case, all the firing degree is considered and the final result
is the sum of the different level of activation (not over one). In any case, the two
methods produce a fuzzy set, which has membership function µ agg ( y ) .
Now we have a result of the fuzzy inference system, which is a fuzzy
replay. We need to return to a “crisp” value, and this step is called
“defuzzification”. This operation produces a “crisp” action y that adequately
represents the membership function µ agg ( y ) . There is no unique way to perform
this operation. To select the proper method, it is necessary to understand the
11
linguistic meaning that underlies the defuzzification process. Two of these
different linguistic meanings are of practical importance: the “best compromise”
and the “most plausible result”. A method of the first type is the Centre of Area
(CoA) that produces the abscissa of the centre of gravity of the fuzzy output set
∫ yµ agg ( y )dy
V
y=
∫ µ agg ( y )dy
V
A method of the second type is the “Mean of Maximum” (MoM). Rather
then balancing out the different inference results, this method selects the typical
value of the terms that is most valid [ ].
Several authors (Chiappero, 2000, Lelli, 2001, Cheli-Lemmi, 1995) have
faced several economic problems using methodological tools based on fuzzy
set theory. Their approach is really different from the one we propose in this first
paper. The unique concept the two approaches have in common is the
necessity to use fuzzy logic and not Boolean logic. The more relevant
differences are two. One is due to the starting point.
These authors start from data they have to pass from a crisp definition to a
fuzzy one of the several concepts they study (functioning, capability, personal
and social characteristics, etc.). Starting with distribution functions they built the
membership functions they need. Next they propose different ways to
aggregate these results to reach the final evaluation.
Here we propose a completely different method. The basic idea is that
this problem is configurable as a multicriteria-problem. The last ones are
faceable with the techniques of Knowledge-Based Systems that is starting from
knowledge that is: experts. The experts describe which are the initial variables
(Var.), how they may be aggregated to have intermediate variables (Int.) and
then how to aggregate the last ones to reach the final evaluation (output). At the
several levels, they use only linguistic attributes and linguistic rules to aggregate
the starting points. They use imprecise information and the process is done
without the knowledge of data. It is a procedure completely torn off data. The
reasoning followed is peculiar of the problem and has not to be affected by
disposable data. They propose clear linguistic attributes and rules. They may be
criticized, but we cannot say that the choices they make are not transparent.
When the system is ready, a sensitive analysis can be done to control if there
are some incongruities. The data are used only at the final step. The data file is
put in the system and the output is ready in a second. The choice of experts
depends on the type of problem we face. In this particular case we think that the
economists are to be supported by psychologists of evolutive age, sociologists
and pedagogists and we will only frame for explicative aims the fuzzy scheme
followed.
Another important difference is from the mathematical point of view. The
unknown function which connects variables with output is, usually, not linear.
Even in case of high non-linearity, these methods are able to approximate it
very well, as we have said before (paragraph 2.1). The mathematical method
other authors propose are average, weighted average and so on. These
methods produce linear functions, but, unfortunately, the real world is too much
12
complex to be linear.
2.2 Structural equation modelling
The existence of multiple, inter-related indicators to measure Children’s Well
being raises the question of how to combine them in empirical research. The
MIMIC model is one approach to this problem.
Confronted with the problem of determining the impact of causes of child well
being, the most basic strategy is to choose a single indicator we believe is the
closest (teenager’s pregnancy for example) to the unobserved construct (child
well being), and ignore both measurement error and information on the
remaining indicators.
Alternately we could use the information in all indicators by creating a synthetic
variable, such as a simple mean indicator. Based upon a set of casual factors
the resulting Ordinary Least Squares model represents perhaps the most
restrictive model given the neglect of measurement error, the reduction of an m
x 1 vector of indicators to a scalar quantity.
In the MIMIC approach we assume that each of the indicators is a component of
child well being; and child well being is an unobserved variable that is linked to
the observable indicators.
The principal advantage of this approach is that it does not rely on exact
measurement of child well being. Each indicator represents a noisy signal of
child well being in a specific age interval.
This modeling strategy has been extensively used in psychometrics and more
recently in econometrics (see for example Raiser, M., Di Tommaso, M.L, and
Weeks M. 2000), and is founded upon the specification of a system of
equations which specify the relationship between a set of unobservable latent
variables, a set of observable endogenous indicators and a set of observable
exogenous variables.
This approach builds upon the early work of Joreskog and Goldeberger (1975)
and Zellner (1970) and has been formalized in the LISREL (Linear Structural
Relationships) model of a set of linear structural equations.
Excellent review of the literature is to be found in Bentler and Weeks (1980) and
Aignes, Hsiao, Kapteyn, and Wansbeek (1984). The Multiple Indicators and
Multiple Causes (MIMIC) approach allows us to think of this model as
comprising two parts: a structural equation for children well being and a
measurement equation that takes into account that there is no single variable
called well-being.
This approach allow us to have an estimate of the influence of a variety of
variables both monetary and non-monetary on child well being at different age
intervals.
13
Section 2.2.1 The MIMIC Approach
In examining the relative merits of our modelling strategy we first introduce
notation. We let
(
Y O = Y1O , Y2O , Y3O ,......... ..YmO
)
'
and Y C = (Y1C , Y2C , Y3C ,...........YmC )
'
denote, respectevily m × 1 vectors of ordinal and continuous indicators;
YiO ∈ {1,....., v}i = 1,.........m, where v represents the number of ordered categories
( )
and Y C represents the latent counterparts to Y O = τ Y C where τ (.) denotes the
one to one mapping between the vector of latent variables and the ordinal
indicators. We will use Y * to denote an unobserved latent construct.
Our argument for choosing the MIMIC specification rests upon the belief that
the parameters, which are delivered by this approach represent the
fundamental objects of interest. In single indicator models, each observed
measure, here elements of the vector Y O , is considered a single indicator of a
matching unobserved construct, elements of Y C , such that the moments of
interest can be written as E (Y O = τ (Y C | x ).
In contrast multiple indicator models (Muthen 1979)) link multiple observed
measures to a reduced dimension of underlying latent variables. In this instance
a single indicator model is not appropriate since the moments we wish to
estimate are of the form E (Y * | Y O = τ (Y C , x ). rather than E (Y O = τ (Y C | x ).
By focussing upon the distinction between these two sets of moments, we can
show the principal differences between a MIMIC specification and more
standard regression-based approaches.
First we note that both MIMIC specification and more standard regressionbased approaches (for example, multivariate probit analogues of SUR models 8)
utilise information in all the m indicators. However, the MIMIC model proposed
8
Multivariate extensions of the binary probit models date from the seminal bivariate probit
model first introduced by Ashford and Sowden(1970). For recent examples see Bock and
Gibbons (1996), and Gibbons (1996), and Golob and Regan (1998).
14
here 9 presupposes the existence of two measurement equations: an inner
equation where each qualitative indicator, say Y jO is linked to its corresponding
continuos counterpart Y jC ; and an outer equation, the standard factor model,
providing a mapping from the multiple indicators collected in Y O to a single
unobserved latent construct, Y * . In contrast discrete versions of SUR, although
admitting dependence across observed indicators, are single indicator
models,and as a result, do not entertain the existence of, for example, an
underlying common factor such as well being.
In this respect the parameters estimated from the set of moments
O
E (Y = τ (Y C | x ). are not the fundamental parameters of interest 10. In our
application we are not interested in the impact of individual characteristics on
each of the dimensions of Well Being.
We have few priors on which to base hypothesis testing with respect to
individual dimensions.
We construct a system of equations, which specify the relationship between a
set of unobservable latent variables Y * , a set of observable endogeneous
ordinal indicators Y O , and a set of observable exogenoeus variables X .
Section 2.2.2 Model Specification
The structure of the model is as follows:
Y = ΛY y * + ε , j = 1,........, m
where
(
Y = Y1 , Y2 , Y3 ,...........Ym
)
'
(1)
9
Given the ordinal nature of our observed measures, the form of the MIMIC model proposed
here is nonlinear and hs been referred to by Wansbeek and Meijer (2000) as the LISCOMP
model.
10
A welcome by-product of the MIMIC approach is that instead of estimating m regression
equations for the set of indicators, we estimate the parameters of a single structural equation.
Ignoring covariance terms, and assuming that x represents a (s x 1)vector of causes, we have a
total of m + s estimable parameters. This compares with a total of m x s parameters and
estimate a system of equations over the m indicators
15
is a m × 1 vector with each element representing an independent indicator of
{
}
children well being, denoted Y * . ΛY = ΛY 1 , ΛY 2 , ΛY 3 .....ΛY j denotes a m × 1
parameter vector of factor loadings, with each element representing the
expected change in the respective indicators following a one unit change in the
latent variable. ε is a m × 1 vector of measurement errors, with Θ ε denote the
covariance matrix.
'
In addition we posit that children’s Well Being is linearly determined by a vector
'
of observable exogeneous variables x = ( x1 , x2 ,.....x s , ) and by a stochastic
error ς giving,
Y * = x 'γ + ς
(2)
where γ is a s ×1 vector of parameters.
Examining (1) and (2) we may think of our model as comprised of two parts: (2)
is the structural (or state) equation and (1) is the measurement equation
reflecting that the observed measurements are imperfect indicators. The
structural equation specifies the casual relationship between the observed
exogeneous causes and children well being. Since Y * is unobserved, it is not
possible to recover direct estimates of the structural parameters γ . Combining
(1) and (2) the reduced form representation is written as
y = πx + v
where π = ΛY γ ' is the m × s reduced form coefficient matrix and v = ΛY ς + ε is the
reduced form disturbance.
For identification issues and treatment of ordinal indicators see Raiser, M., Di
Tommaso, M.L, and Weeks M. 2000
16
Section 3 – An application to the capability of social interaction.
Section 3.1 Building a System of indicators to measure the Capability of
Social Interaction
As stated in the first part of this paper the capability of social interaction and its
functioning can be affected by several environmental factors that we will try to
highlight in this section aiming also at an application to Italian data.
The very possibility to interact depends on the presence of other individual in
the environment where the child lives, therefore it is important to have data at
aggregate level on fertility rates and on average number of children in the area
where the child lives and it would be better to have access to individual data
allowing to get precise information on the household’s composition of the child,
as well as on the employment condition of parents and their education. One
should be aware that the same individual characteristic may affect social
interaction capability in different ways. For instance a parent being manager
may have only a limited amount of time to devote to interacting with the child
(this should have a negative effect on functioning parent’s interaction with child)
but can increase family income and thus can make affordable private childcare
services or access to children laboratories or to a bundle of commodities and
services that may allow the child a higher level of interaction with other children
or adults.
One should also analyse the impact of institutional factors (laws, collective
agreements) on the employed parents’ possibility to interact with their child. On
this regards Law 53/2000 in Italy made it possible for parents to use parental
leave up to the child’s age of 8. By using ISTAT 2002-2003 survey on time use,
ISTAT (2004) shows that if more than 2/5 of employee working women used
this possibility amongst the 44,000 working mothers who did not use the leave
5% (especially blue collars) stated that they were not allowed to use it.
Having access to extended family composition data could provide information
also on the probability of having cousins and other relatives that the child can
meet (therefore data on how often does the child meet relatives of different age
is useful).
Another area where the child can experience social interaction is at school or
kindergarten and it becomes significant to test at each age level the diffusion of
child care services and schools of different types in the area where the child
lives. One should notice that the presence of childcare services for children
under 3 years old is particularly low in Italy and not evenly distributed across the
different regions as Tab.1 and 2 from Istituto degli Innocenti (2002) clearly
show. Also full-time school is not evenly distributed across Italy, nor the
possibility of having lunch at school with other school mates.
17
Tab. 1 Percentage of places available in
kindergarten (private and public)
for region in Italy on children
aged 0-2 living in each region
Piemonte
Valle d'Aosta
Lombardia
Trentino Alto Adige
Veneto
Friuli Venezia Giulia
Liguria
Emilia Romagna
Toscana
Umbria
Marche
Lazio
Abruzzo
Molise
Campania
Puglia
Basilicata
Calabria
Sicilia
Sardegna
Italy
Source Istituto degli Innocenti (2002)
tav.3 p.17
%
10,7
12,3
9,7
7,5
7,2
7,8
9,7
18,3
11,3
11,6
11,5
8,5
4,1
2,9
2,2
2,7
5,2
1,9
4,7
6,4
7,4
18
Tab. 2 Percentage of places in public kindergarten
available for region in Italy on children
aged 0-2 living in each region
%
Piemonte
9,7
Valle d'Aosta
12,3
Lombardia
8,9
Bolzano
3
Trento
11
Veneto
5,5
Friuli Venezia Giulia
6,1
Liguria
9,3
Emilia Romagna
17,4
Toscana
10,7
Umbria
10,6
Marche
9,7
Lazio
7,5
Abruzzo
3,8
Molise
2,5
Campania
1
Puglia
2,1
Basilicata
4,4
Calabria
1,1
Sicilia
4,7
Sardegna
5,7
Italy
6,5
Source Istituto degli Innocenti (2002)
New types of services are spreading in Italy for children in preschool age and
could be particularly useful to increase social interaction with other children
especially for those who could not have access to kindergarten, they are
unevenly present in different regions and only 0,62% of children aged less than
3 do attend them on average in Italy.
It could be useful to have data also on quality of childcare,11 since time spent in
high quality childcare services has been found to have a positive impact on
child’s social competence an cooperation later in his life and on peer relations. 12
The capability of social interaction can be realized by interacting with other
children during courses or laboratories or in open spaces (therefore it becomes
relevant at an aggregate level to measure the supply of laboratories or societies
caring for the diffusion of these activities and the presence of parks in the areas
11
On the indicators of quality of childcare see Waldfogel (2002). As she stresses: ‘…there is no
consensus on how to define quality of child care. Child care advocates tend to point to structural
features of child care programs such as the group size, child-staff ratio, and health and safety
requirements, while parents tend to look for a caregiver who is warm and sensitive, and
conveniently located. Researchers try to measure both types of characteristics, as well as
continuity and stability of care’ (Waldfogel, 2002, p.544).
12
These effects have been found in different studies based on US and UK data referred to by
Waldfogel (2002 pp.540-541).
19
where the child lives, whereas at micro level it is important to measure the
attendance of children of these spaces and activities).
Measuring functionings of this capability involves having individual data on how
often the child meets other children, relatives or other people and it is important
to see what type of activities they do together (for instance spending 3 hours a
day with a brother watching television or making videogames is different than
having a football match in the open air). In a gender perspective it is also
important to see how often a child meets persons of different sex.
We will try to make an application to measuring social capability for children
aged from 6 to 10 living in Italy by introducing the flow in Figure 1 and by
implementing it by means of different models in the following sections.
Number of times the
child interacts with
brothers and/or sisters
Household’s
sociodemographic
composition
Interaction with schoolmates
Interaction with teachers
Interaction with other
children or adults in
social activities
Extended family composition
Presence of full-time
elementary school and of
laboratories, possibility to
have lunch at school
Social interaction
Interaction with
grandparents and
other adult relatives
Diffusion of profit and non profit
institutions providing laboratories
in arts, gym or other social
activities
Interaction with
cousins
Interaction with
mother
Interaction with
father
Fig.1 Social interaction, functionings and environmental factors.
This flow on the left hand side assumes environmental factors affecting
capability of social interaction as well as its conversion in functionings which are
listed in the right hand side of the flow.
20
Section 3.2 Building a Fuzzy set model to measure the Capability of Social
Interaction
In this section we will try to apply the logical system of indicators presented in
Section 3.1 by looking at useful data and by proposing a fuzzy expert scheme to
measure them. The main source of data that we will use is ISTAT (Italian
Institute for Statistics) 1998 Famiglia, soggetti sociali e condizione dell’infanzia
survey on about 24,000 households and about 60,000 individuals that contains
together with useful information on each child’s family composition (though not
household’s income) also information on each child’s variables (sex, age, level
and type of education) and on each child’s activities with relatives and other
individuals.
The proposed scheme can disaggregate the variables affecting social
interaction capability as following:
Individual information affecting social interaction:
-
-
Household socio-demographic composition:
- Number of member and age
- Household type: one earner, double earner, lone parent
- Parents’ education level and their employment condition
Extended household composition: presence of grandparents and number of
cousins
Attendance of full-time elementary school (hours at school)
Possibility to have lunch at school
Hours spent in front of the television (this can limit the total amount of time
devoting to other activities and to interact with others)
Whether the child has experienced during last year any violent act from
other children of the same age
How often does the child play in the open air
Aggregate data on the presence of full time elementary school, or on the
attendance of laboratories in the region13 where the child lives can be obtained
by ISTAT multipurpose survey taking into account also the size of town where
the family lives.
Observed functionings can be aggregated into:
-
-
13
Interaction with brother or sisters (whether the child sleep alone or with
brother/sister; whether the child returns or go to school with brother or
sister, whether the child plays with brother or sisters)
Interaction with schoolmates (whether the child plays with schoolmates)
Interaction with cousins (whether the child meets them once or more
than once a month)
The survey is statistically significant at regional level.
21
-
-
-
-
-
Interaction with friends (whether the child returns or go to school with
friends, whether the child plays with friends meeting them outside
structured activities and how often does the child meet them, whether the
child meet more children of his/her sex, whether the child participates at
parties)
Interaction with grandparents (whether the child returns or go to school
with grandparents, whether the child is cared by his/her grandparents
while not at school, whether the child plays with grandparents)
Interaction with mother (whether the child returns or go to school with
her, activities that the child does with her, how often does the child play
with her, whether the child sleeps with her)
Interaction with father (whether the child returns or go to school with him,
activities that the child does with him, how often does the child play with
him, whether the child sleeps with him)
Interaction with other in cultural, environmental or ‘boy scouts’ societies
(whether the child participates and how often)
Interaction with other children or adults in social structured activities
(gym, music, arts…): whether the child attends courses (paid or unpaid)
how many times a week (the survey provide us information on the
number of hours devoted by each child to each course)
By using fuzzy set we can measure each observed functioning (by aggregating
different indicators of the same functioning according to rule that will be made
explicit) and then aggregate the functionings to have an estimate of the
unobserved child’s capability of social interaction.
To make a first attempt we can apply fuzzy expert system to two functionings:
interaction with father and interaction with other children or adults in social
structured activities (gym, music, arts…):
Interaction with father
For this functioning we can observe:
How often does he play:
If he plays with the father every day or sometimes during the week is good for
social interaction, if he plays once a week is not so bad, if he plays less than 4
times a month or a few times in the year is bad and if he never plays is very
bad.
Types of games made more frequently with father:
Bad if videogames, good if others (games of role, puzzles, in gender
perspective it can be very good if they play with the father by performing
housework activities cooking and so on)
Observed variables affecting the conversion of social interaction in this specific
functioning are:
- Father’s presence
- Father’s education level
- Father’s employment condition
22
Interaction with other children or adults in social structured activities
For this functioning we can observe whether child attends courses (paid or
unpaid) and how many times a week (the survey provide us information on the
number of hours devoted by each child to each course)
Observed variables affecting this functioning:
This variable is affected by family income and by the public and private (profit
and non profit) supply of these courses that one should try to measure on
average for each region.
Here we propose the Fuzzy scheme of “Interaction with father” (output)
For this functioning we can say that the variables affecting the conversion of
social interaction in this specific functioning are:
- Father’s presence (var1)
- Father’s education level (var2)
- Father’s employment condition (var3)
How often does he play (Var4)
Types of games made more frequently with father (var5).
The first three variables are aggregated together in an intermediate variable
(int1) we may call “Father”.
Then Int1, var4 and var5 are aggregated together to obtain the final otuput as in
Fig.2.
Fig. 2 Interaction with father – a fuzzy representation
We may begin with three linguistic attributes for every variable1-5: Low,
Medium, High.
As example a way to fuzzify the var4 may be this type of judgement:
If he/she plays with the father every day or sometimes during the week is good
for social interaction, if he plays once a week is not so bad, if he plays less than
4 times a month or a few times in the year is bad.
A way to fuzzify the var5 may be this type of judgement:
23
Bad if videogames, good if others (games of role, puzzles), in gender
perspective it can be very good if they play with the father by performing
housework activities cooking and so on.
The intermediate variable Int1 has usually a more detailed type of judgement
with five linguistic attributes: Very Low, Low, Medium, High, Very High.
Combining the three observed variables may give rise to an outcome of the type
described in Tab.3.
IF
var1
low
low
low
low
low
low
low
low
low
medium
medium
medium
medium
medium
medium
medium
medium
medium
high
high
high
high
high
high
High
High
High
var2
low
low
low
medium
medium
medium
high
high
high
low
low
low
medium
medium
medium
high
high
high
low
low
low
medium
medium
medium
high
high
high
var3
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
DoS
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
THEN
int1
very_low
very_low
low
very_low
low
medium
low
medium
high
very_low
low
medium
low
medium
high
medium
high
very_high
low
medium
high
medium
high
very_high
high
very_high
very_high
Table 3: Rules of the Rule Block "Father"
We can then combine the results of the first block of variables with variables
4 (how often does the father play with child) and 5 (what types of games are
more frequently made with father) to get the outcome for ‘interaction with
father’.
24
IF
int1
very_low
very_low
very_low
very_low
very_low
very_low
very_low
very_low
very_low
low
low
low
low
low
low
low
low
low
medium
medium
medium
medium
medium
medium
medium
medium
medium
high
high
high
high
high
high
high
high
high
very_high
very_high
very_high
very_high
very_high
very_high
very_high
very_high
very_high
var4
low
low
low
medium
medium
medium
high
high
high
low
low
low
medium
medium
medium
high
high
high
low
low
low
medium
medium
medium
high
high
high
low
low
low
medium
medium
medium
high
high
high
low
low
low
medium
medium
medium
high
high
high
var5
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
low
medium
high
DoS
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
THEN
output
very_low
very_low
very_low
low
low
low
medium
medium
medium
low
low
low
medium
medium
medium
medium
medium
medium
low
low
low
medium
medium
medium
high
high
high
medium
medium
medium
medium
medium
medium
high
high
high
medium
medium
medium
high
high
high
very_high
very_high
very_high
Table 4: Rules of the Rule Block “Interaction_with_father”
However this is only a first attempt that we would like to fill with ‘sense’. For
instance we can have a psychologist’s view on how the type of game the father
25
does with the child can be judged (the rule that we have expressed in this
simulation is just based on our personal judgement) or expert’s view on the
effect of father’s employment status or education on the capability of interaction
with his child.
Moreover the same environmental variable can have a different impact on
different functionings of the same capability. For instance having a manager as
father can imply having a father with a limited amount of leisure that he can
devote to interaction with his child (negative effect on interaction with father) but
can be a proxy of a higher family income that can allow the child to improve
his/her social interaction in paid courses.
4. Concluding remarks and future research
This paper explores the possibilities of using fuzzy inference system and
structural equation modelling to measure capabilities referred to children’s well
being both at a theoretical and empirical level.
In this paper we present a first attempt to use a fuzzy expert system (FES) to
face the child well being. The necessity to use a fuzzy approach, felt by other
authors, here is treated in a new way. We utilize an instrument that is typical of
Artificial Intelligence, in which the support of experts, economists and
psychologists is a fundamental tool. As this fuzzy support has had a lot of
successful applications, not only in engineering world but even in economic and
financial ones, (Facchinetti G.- Cosma S.- Mastroleo G.- Ferretti R. (2001),
Facchinetti G.- Franci F.-Mastroleo G.- Pagliaro V.- Ricci G. (2002), Facchinetti
G. -Lalla M. -Mastroleo G. (2003), Facchinetti G.- Mannino I.- Mastroleo G.Soriani S. (2003), Magni C.A. - Mastroleo G. - Vignola M. - Facchinetti, G.
(2004)), we trust that it is a good proposal even in this case and in several
others connected, like quality of life, of work and so on. The attempt to apply
this method to children’s well being is only a preliminary attempt, before filling it
with available data, we think it is necessary to enlarge our research group to
experts from other disciplines that can provide theoretical reasonings for the
rules that we will assume in the fuzzy scheme. Our aim is to insert experts’
justified rules in the fuzzy scheme that we are building and to highlight possible
interactions with different environmental variables and capabilities leading to
child well being in Italy by using available data set. The scheme we are
currently working to can allow for simulation of different theoretical views on the
builiding of children’s capabilities, as well as simulations on how children living
in different areas or having different household structures have different well
being and different functionings by entering in the same fuzzy set scheme
values for functionings referring to children from different areas or different
household types currently available.
We will then analyze how the different techniques used to measure child well
being differ according to the obtained results and theoretical expectations.
26
Bibliography
-
-
-
-
-
-
-
-
-
Aigner, D. J., C. Hsiao, A. Kapteyn, and T. Wansbeek (1984): “ Latent
Variable Models in Econometrics” in Handbook in Econometrics, ed. by Z.
Griliches, and M.D. Intriligator , vol. II, pp 1323-1393. North Holland,
Amsterdam.
Bentler, P.M., and D.G. Weeks (1980): “ Multivariate Analysis with Latent
Variables” in Handbook of Statistics, ed. by P.R. Krishnaiah, and L.Kanal, pp
747-771, North Holland, Amsterdam.
Cheli, B. – Lemmi, A. (1995) ‘A totally fuzzy and relative approach to the
multidimensional analysis of poverty’, Economic Notes, 1:115-134.
Chiappero Martinetti E. 1996. Standard of living evaluation based on Sen's
approach: some methodological suggestions, Notizie di Politeia 12(43/44):
37-53.
Chiappero Martinetti E. 2000. "A Multidimensional Assessment of WellBeing Based on Sen's Functioning Approach." Rivista Internazionale di
Scienze Sociali 108:207-39.
Di Tommaso M.L., 2003, “How to Measure Children Wellbeing Using a
Capability Approach? An Application to Indian Data through a Multiple
Indicators
Multiple
Causes
Model”
available
at:
http://cfs.unipv.it/sen/program.htm
Facchinetti G.- Franci F.-Mastroleo G.- Pagliaro V.- Ricci G. (2002). ”From a
logic map to a fuzzy expert system for the description of the Middle East
destabilization”. Presented as Plenary Session in the 9th International
Conference on ADVANCED COMPUTER SYSTEMS, ACS'2002 October
23-25, 2002 Miedzyzdroje, Poland. Published in Proceedings of ACS 2002,
Soldek J., Drobiazgiewicz L. Editors. Part I, 27-36. ISBN 83-87362-47-6.
Facchinetti G. -Lalla M. -Mastroleo G. (2003): “Ordinal Scales and Fuzzy Set
System to Measure Agreement: An Application to the Evaluation of
Teaching Activity”. Quality and Quantity accepted for printing.
Facchinetti G.- Mannino I.- Mastroleo G.- Soriani S.: (2003) “A fuzzy expert
approach for comparing alternative end uses for requalification of
contaminated sites”. Plenary Session in the 10th International Conference
on ADVANCED COMPUTER SYSTEMS, ACS' 2003 October 23-25, 2003
Miedzyzdroje, Poland. Published in Proceedings of ACS 2003, Soldek J.,
Drobiazgiewicz L. Editors
Facchinetti G.-Cosma S.-Mastroleo G.–Ferretti R. (2001) A fuzzy credit
rating approach for small firm creditworthiness evaluation in bank lending.
An Italian case” Proceedings del CIMA 2001, International ICSC-NAISO
Congress on Computational Intelligence: Methods & Applications, June 1922 2001 Bangor, Wales, U.K. ISBN: 3 -906454-26-6.
ISTAT (2004) Rapporto Annuale 2003, ISTAT.
Istituto degli Innocenti (2002) ‘I servizi educativi per la prima infanzia.
Indagine sui nidi d’infanzia e sui servizi educativi 0-3 anni integrativi al nido
al 30 settembre 2002’ Quaderni del Centro Nazionale di documentazione e
analisi per l’infanzia e l’adolescenza. www.minori.it
27
-
-
-
-
-
Joreskog, K. G., and A. S. Goldberger (1975), “Estimation of a Model with
Multiple Indicators and Multiple Causes of a Single Latent Variable,” Journal
of the American Statistical Association, 70, 631-639.
Kosko, B. (1992). Fuzzy Systems as Universal Approximators. Proc. IEEE
Int. Conf. On Fuzzy Systems, 1153-1162.
Lelli, S. (2001), "Factor Analysis vs. Fuzzy Sets Theory: Assessing the
Influence of Different Techniques on Sen's Functioning Approach", CES
Discussion Paper Series, 01.21, Katholieke Universiteit Leuven.
Magni C.A. - Mastroleo G. - Vignola M. - Facchinetti, G. (2004) “Strategic
Options and Expert Systems: A Fruitful Marriage“. Soft Computing Vol 8,
n°.3, 2004 179-192
Nussbaum M. C. (2003) “Capabilities as Fundamental Entitlements: Sen and
Social Justice, Feminist Economics,(2).
Raiser M., Di Tommaso, M.L. Weeks M. (2000), A Structural Model of
Institutional Change: Evidence from Transition Economies, Working Paper
of the Dep. of Applied Economics, University of Cambridge.
Robeyns I. (2003), “Sen’s Capability Approach and Gender Inequality:
Selecting Relevant Capabilities” Feminist Eocnomics, vol 9 (2-3), pp 61-92.
Von Altrock, C. (1997) Fuzzy logic and Neurofuzzy applications in Business
and Finance, Prentice-Hall, Inc
Waldfogel, J. (2002) ‘Child care, women’s employment, and child outcomes’
Journal of Population Economics 15:527-548.
Wang L.(1992), “Fuzzy systems are universal approximators. Proc. Of Int.
Conf. On Fuzzy Engineering”, 471-496.
Wansbeek T. and Meijer E., (2000), Measurement Error and Latent
Variables in Econometrics, North Holland, Elsevier Science.
Zadeh L.A. (1965) “Fuzzy sets”. Information and Control. Vol 8 pp. 338-353.
Zellner, A (1970), “Estimation of Regression Relationships Containing
Unobservable Variables,” International Economic Review, 11, 441-454.
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